3D Koch-type crystals

Abstract

We consider the construction of a family \{K_N\} of 3 -dimensional Koch-type surfaces, with a corresponding family of 3 -dimensional Koch-type “snowflake analogues” \{\mathcal{C}_N\} , where N>1 are integers with N \not\equiv 0 \mathrm{mod}\,\,3 . We first establish that the Koch surfaces K_N are s_N -sets with respect to the s_N -dimensional Hausdorff measure, for s_N=\log(N^2+2)/\log(N) the Hausdorff dimension of each Koch-type surface K_N . Using self-similarity, one deduces that the same result holds for each Koch-type crystal \mathcal{C}_N . We then develop lower and upper approximation monotonic sequences converging to the s_N -dimensional Hausdorff measure on each Koch-type surface K_N , and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set \mathcal{C}_N . As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals \mathcal{C}_N , for N>2 .